A derivation of the master equation from path entropy maximization

Abstract

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive n-th order Markov processes and the master equation as unique solutions to an inverse problem. In particular, we find that when the constraints are not enough to uniquely determine the stochastic model, the n-th order Markov process emerges as the unique maximum entropy solution to this otherwise under-determined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from first principles.